In mathematics, a radially unbounded function is a function for which [1]
Or equivalently,
Such functions are applied in control theory and required in optimization for determination of compact spaces.
Notice that the norm used in the definition can be any norm defined on , and that the behavior of the function along the axes does not necessarily reveal that it is radially unbounded or not; i.e. to be radially unbounded the condition must be verified along any path that results in:
For example, the functions
are not radially unbounded since along the line , the condition is not verified even though the second function is globally positive definite.